A simple random sample of size n equals 10 is obtained from a population with mu equals 66 and sigma equals 16.
What must be true regarding the distribution of the population?
A.Since the sample size is large enough, the population distribution does not need to be normal.
B. The population must be normally distributed and the sample size must be large.
C. The population must be normally distributed and the sample size must be large.
D.The sampling distribution must be assumed to be normal.
B. The population must be normally distributed and the sample size must be large.
In order to use statistical methods that assume a normal distribution, such as confidence intervals or hypothesis tests, the population from which the sample is obtained must be normally distributed. Additionally, for the Central Limit Theorem to apply and for the sampling distribution to be approximately normal, the sample size should be large (typically greater than 30).
To answer this question, we need to understand the concept of sampling and its relationship with the population distribution.
In statistical sampling, we select a sample from a larger population to make inferences or draw conclusions about the entire population. The accuracy of our inferences depends on the characteristics of both the sample and the population distribution.
Given that we have obtained a simple random sample of size 10, we can first determine whether the sample size is considered large enough. Generally, if the sample size is large enough, we can assume that the distribution of the sample will be approximately normal, regardless of the population distribution. However, there is no specific rule that defines what constitutes a "large" sample size.
Based on the information provided, we don't have any evidence regarding the population distribution. We only know the mean (μ = 66) and standard deviation (σ = 16) of the population. Therefore, we cannot conclude with certainty that the population must be normally distributed.
Option A states that the population distribution does not need to be normal because the sample size is large enough. This is a possible answer if we consider the sample size as large enough.
Option B states that the population must be normally distributed, which is incorrect based on the information given. The sample size must also be large, which is not a requirement for assuming normality.
Option C states the same incorrect information as option B.
Option D states that the sampling distribution must be assumed to be normal. This is incorrect as well, as we have no information about the sampling distribution.
Based on the information provided, the correct answer is A. Since the sample size is not specified as large, we cannot conclude anything about the population distribution.